why natural phenomena follow exponential growth?

1: In the context of the definition given below, what does the word "shallower" mean? Here is given the exponential graph.

exponential
Something is said to increase or decrease exponentially if its rate of change must be expressed using exponents. A graph of such a rate would appear not as a straight line, but as a curve that continually becomes steeper or shallower.

The American Heritage Science Dictionary

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2: I have read that many phenomena in nature follow exponential growth. Why is so? Could you please explain how this connection is established between nature and mathematical world? Obviously, if bacterial follows exponential growth, then it is not under any obligation to do so, it does it unknowingly. Besides exponential growth, what other type of growth the natural phenomena most often follow? Please don't use more math to explain math. Thank you.

Look at the population of the human race since the industrial revolution. Since we have successfully eliminated most of nature's checks & balances for population control, our own growth has been exponential.

given food and safety, the reproduction of just about any species will be exponential.

Look at the population of the human race since the industrial revolution. Since we have successfully eliminated most of nature's checks & balances for population control, our own growth has been exponential.

given food and safety, the reproduction of just about any species will be exponential.

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Thank you, Strantor, THE RB.

1: I have checked the definitions of "shallow" in Merriam Webster and American Heritage and nowhere it suggests that "shallow" in any way carries meaning opposite to "steep". Please help me with it.

exponential
Something is said to increase or decrease exponentially if its rate of change must be expressed using exponents. A graph of such a rate would appear not as a straight line, but as a curve that continually becomes steeper or shallower.

The American Heritage Science Dictionary

Click to expand...

2: It is said that human population growth rate is also exponential. But I don't see it. At least where I live some people have one kid, some two, some as many as five, and so on. I think people don't really care much about exponential growth! What's your opinion on this?

1: I have checked the definitions of "shallow" in Merriam Webster and American Heritage and nowhere it suggests that "shallow" in any way carries meaning opposite to "steep". Please help me with it.

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Since English is not your first language and you had to learn it, then no doubt you have realized the idiosyncrasies and double standards in it that native speakers take for granted. You could write a book about how much English doesn't make sense. I couldn't think of a word that is the opposite of steeper, so I googled and I'm not the only one. I guess whoever wrote that definition in your first post couldn't come up with a good word either, so they used "shallower" - even though it doesn't quite fit. what they meant was "a curve that continually becomes steeper or (opposite of steeper)"

2: It is said that human population growth rate is also exponential. But I don't see it. At least where I live some people have one kid, some two, some as many as five, and so on. I think people don't really care much about exponential growth! What's your opinion on this?

The opposite of steeper could be flatter, and I don't mean flatter as in give compliments and praise. It's weird because the opposite of steep isn't flat because flat would be the opposite of vertical. But, shallow seems acceptable to me in this context. I've heard the term shallow grade or shallow roof, referring to a nearly flat slope.

You see, most biologic systems, from bacterial populations to predator-pray equilibriums, can be described very closely by a model of differential equations.
This kind of equations usually have an exponential solution and thus its dominance when describing those systems.

So, in a sense, it's the way we describe those systems that leads to the dominance of the exponential curve.

Check some basic differential books. They have many examples of natural systems. I think that book has some of them:

2: I have read that many phenomena in nature follow exponential growth. Why is so? Could you please explain how this connection is established between nature and mathematical world? Obviously, if bacterial follows exponential growth, then it is not under any obligation to do so, it does it unknowingly. Besides exponential growth, what other type of growth the natural phenomena most often follow? Please don't use more math to explain math.

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PG,

It seems you've gotten examples of exponential growth (or decay), but no explanation why these happen in nature without nature having a knowledge of 'exponential'. The answer is pretty easy:

In general, the odds of something happening in a given population of 'things' is a linear function of the number of 'things' in the population. The instantaneous rate-of-change of the population is, for example, some constant times the instantaneous size of the population. Assuming constant is positive (exponential growth), each 'generation' grows the overall size of the population (at the rate-of-change), therefore, the rate-of-change also increases. It is the increasing (or decreasing) rate-of-change of population with respect to the size of the population that defines the exponential growth or decay. But it is all based upon a simple *linear* function:

rate-of-change = constant * population-size

The constant can be positive (exponential growth) or negative (exponetial decay). For example, think of the half-life of an atomic isotope:

Tritium has a half life of ~12 years. This means that for any given quantity of tritium, in 12 years half of it will be converted to something else, and half will remain.

12 years later, yet another half will be converted, and so on.

This is exponential decay:

Year Tritium Remaining
0 100%
12 50%
24 25%
36 12.5%

and so on.

Keep in mind that the 'constant' may not be constant at all. It may also change with respect to population density, energy (food) availability, waste accumulation, or other things. For example, with respect to bacteria populations, the growth is fast early on (many bacteria divide, few die). Then, as the population increases, demand for food increase, and waste products are not as easily eliminated. The population then stabilizes. Finally, the food may be exhausted, or they my poison themselves with their own waste, and the population dies.

Each phase of the life cycle above is simply variations on the constant involved, going from a positive value, to zero, to a negative value.

EDIT: Misspoke above: The odds are generally constant regardless of population size...the number of times something actually happens is a linear function of the population size and is defined by odds*population_size.

The notion of shallow and steep is pretty common when discussing slopes of curves. The dividing point is a slope of around 1; slopes greater than that are considered steep (with increasing slope meaning "steeper"); conversely, slopes less than 1 are considered shallow, with decreasing slope implying "shallower". The usage is analogous to the common usage of "shallow" and "steep" when e.g. discussing hiking terrain.

The notion of shallow and steep is pretty common when discussing slopes of curves. The dividing point is a slope of around 1; slopes greater than that are considered steep (with increasing slope meaning "steeper"); conversely, slopes less than 1 are considered shallow, with decreasing slope implying "shallower". The usage is analogous to the common usage of "shallow" and "steep" when e.g. discussing hiking terrain.

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Well, I learn something new every day. I've never heard shallower used that way, even talking about hiking slopes. Until now I've only heard it used in reference to decreasing depth of water (which would technically be an increasing slope). I could get used to it if I heard it every day, but for now it doesn't sound right to my ear.